Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc (Q2839306)

Let NEWLINENEWLINENEWLINE\[NEWLINE\mathbf{s}=(\varphi_1,\dots,\varphi_n):\mathbb C^n\longrightarrow\mathbb C^n,\quad \varphi_i(z_1,\dots,z_n):=\sum_{1\leq k_1<\dots<k_i\leq n}z_{k_1}\cdots z_{k_i}NEWLINE\]NEWLINE NEWLINENEWLINEand let \(\mathbb G_n:=\mathbf{s}(\mathbb D^n)\), where \(\mathbb D\) stands for the unit disc. The domain \(\mathbb G_n\) is called the symmetrized polydisc. For \(\lambda>1\) let \(dV^{(\lambda)}\) be the probability measureNEWLINENEWLINENEWLINE\[NEWLINE\bigg(\frac{\lambda-1}{\pi}\bigg)^n\prod_{i=1}^n(1-r_i^2)^{\lambda-2}r_idr_id\theta_i\quad\text{on}\quad \mathbb D^n,NEWLINE\]NEWLINE and let \(dV_{\mathbf{s}}^{(\lambda)}\) be the measure on \(\mathbb G_n\) such that NEWLINE\[NEWLINE\int_{\mathbb G_n}fdV_{\mathbf{s}}^{(\lambda)}=\int_{\mathbb D^n}(f\circ\mathbf{s})|J_{\mathbf{s}}|^2dV^{(\lambda)},NEWLINE\]NEWLINE where \(J_{\mathbf{s}}(z)=\prod_{1\leq i<j\leq n}(z_i-z_j)\) is the complex Jacobian of \(\mathbf{s}\). Let NEWLINENEWLINENEWLINE\[NEWLINE\mathbb A^{(\lambda)}(\mathbb G_n):=\mathcal O(\mathbb G_n)\cap L^2\big(\mathbb G_n,dV_{\mathbf{s}}^{(\lambda)}\big)NEWLINE\]NEWLINE be the weighted Bergman space and let NEWLINENEWLINENEWLINE\[NEWLINE\mathbb A^{(\lambda)}_{\text{anti}}(\mathbb D^n):=\Big\{f\in\mathcal O(\mathbb D^n)\cap L^2\big(\mathbb D^n,dV^{(\lambda)}\big)\;:\, f \text{ is anti-symmetric}\Big\}.NEWLINE\]NEWLINE Using the isomorphism of the Hilbert spaces \(\mathbb A^{(\lambda)}(\mathbb G_n)\) and \(\mathbb A^{(\lambda)}_{\text{anti}}(\mathbb D^n)\), the authors obtain the following effective formula for the reproducing kernel for \(\mathbb A^{(\lambda)}(\mathbb G_n)\): NEWLINE\[NEWLINE \mathbf{B}_{\mathbb G_n}^{(\lambda)}\big(\mathbf{s}(z),\mathbf{s}(w)\big)=\frac{\|J_{\mathbf{s}}\|^2_{L^2(\mathbb D^n, dV^{(\lambda)})}\det\big((1-z_j\overline w_k)^{-\lambda}\big)_{j,k=1}^n}{n!a(z)\overline{a(w)}},\quad z,w\in\mathbb D^n, NEWLINE\]NEWLINE where \(a(\xi):=\det((\xi_i^{n-j})_{i,j=1}^n)\). Moreover, in the second part of the paper, the authors prove the following formula for the Szegő kernel of \(\mathbb G_n\): NEWLINE\[NEWLINE \mathbb S_{\mathbb G_n}(\mathbf{s}(z),\mathbf{s}(w))=\prod_{j,k=1}^n\frac1{1-z_j\overline w_k},\quad z,w\in\mathbb D^n. NEWLINE\]